Marco D'Anna

An amalgamated duplication of a ring along an ideal: the basic properties

We introduce a new general construction, denoted by R ⋈ E, called the amalgamated duplication of a ring R along an R-module E, that we assume to be an ideal in some overring of R. (Note that, when E2 = 0, R ⋈ E coincides with the Nagatas idealization R ⋉ E.). After discussing the main properties of the amalgamated duplication R ⋈ E in relation with pullback-type constructions, we restrict our investigation to the study of R ⋈ E when E is an ideal of R. Special attention is devoted to the ideal-theoretic properties of R ⋈ E and to the topological structure of its prime spectrum.

Analytically unramified one-dimensional semilocal rings and their value semigroups

Abstract In a one-dimensional local ring R with finite integral closure each nonzerodivisor has a value in N d , where d is the number of maximal ideals in the integral closure. The set of values constitutes a semigroup, the value semigroup of R. We investigate the connection between the value semigroup and the ring. There is a particularly close connection for some classes of rings, e.g. Gorenstein rings, Arf rings, and rings of small multiplicity. In many respects, the Arf rings and the Gorenstein rings turn out to be opposite extremes. We give applications to overrings, intersection numbers, and multiplicity sequences in the blow-up sequences studied by Lipman.

A Family of Quotients of the Rees Algebra

A family of quotient rings of the Rees algebra associated to a commutative ring is studied. This family generalizes both the classical concept of idealization by Nagata and a more recent concept, the amalgamated duplication of a ring. It is shown that several properties of the rings of this family do not depend on the particular member.

INVARIANTS OF IDEALS HAVING PRINCIPAL REDUCTIONS

For a regular ideal having a principal reduction in a Noetherian ring we consider the structural numbers that arise from taking the Ratliff–Rush closure of the ideal and its powers. In particular, we analyze the interconnections among these numbers and the relation type and reduction number of the ideal. We prove that certain inequalites hold in general among these invariants, while for ideals contained in the conductor of the integral closure of the ring we obtain sharper results that do not hold in general. We provide applications to the one-dimensional local setting and present a sequence of examples in this context.

On the Buchsbaumness of the Associated Graded Ring of a One-Dimensional Local Ring

Let (R, m) be a Noetherian, one-dimensional, local ring, with |R/m|=∞. We study when its associated graded ring G(m) is Buchsbaum; in particular, we give a theoretical characterization for G(m) to be Buchsbaum not Cohen–Macaulay. Finally, we consider the particular case of R being the semigroup ring associated to a numerical semigroup S: we introduce some invariants of S, and we use them in order to give a necessary and a sufficient condition for G(m) to be Buchsbaum.

Normal Hilbert functions of one-dimensional local rings

Let (R, m) be a one-dimensional reduced (Noetherian) local ring with finite integral closure ( , M 1,…,M t ). We assume further that R/m ≃ /M i for each i and that Card(R/m) ≥ t. We study for such a ring R the associated graded ring and the Hilbert series, with respect to the normal filtration of an m-primary ideal I, R ⊇ . We make use of the value semigroup of R and in particular of some results of [7].

GOOD SUBSEMIGROUPS OF ℕn

Value semigroups of non-irreducible singular algebraic curves and their fractional ideals are submonoids of ℤn that are closed under infimums, have a conductor and fulfill a special compatibility property on their elements. Monoids of ℕn fulfilling these three conditions are known in the literature as good semigroups and there are examples of good semigroups that are not realizable as the value semigroup of an algebraic curve. In this paper, we consider good semigroups independently from their algebraic counterpart, in a purely combinatorial setting. We define the concept of good system of generators, and we show that minimal good systems of generators are unique. Moreover, we give a constructive way to compute the canonical ideal and the Arf closure of a good subsemigroup when n = 2.